Asymptotic Efficiency of a Class of Non-Parametric Tests for Regression Parameters

Abstract

For testing hypotheses about $\alpha$ and $\beta$ in the linear regression model $Y_j = \alpha + \beta x_j + Z_j$, Brown and Mood [18] have proposed distribution-free tests, based on their median estimates. Daniels [6] has also given a distribution-free test for the hypothesis that the regression parameters have specified values. This latter test is an improvement on the Brown and Mood median procedure, although both are based on the signs of the observations. Recently Hajek [10] constructed rank tests, which are asymptotically most powerful, for testing the hypothesis that $\beta = 0$, while $\alpha$ is regarded as a nuisance parameter. In this paper, a class of rank score tests for the hypothesis $H : \alpha = \beta = 0$, is proposed in Section 2. This class includes as special cases, the Wilcoxon and the normal scores type of tests. In Sections 3 and 4 the limiting distribution of the test statistics is shown to be central $\chi^2$, under $H$, and non-central $\chi^2$, under a sequence of alternatives tending to the hypothesis at a suitable rate. In Section 5, the Pitman efficiency of the proposed tests relative to the classical $F$-test, is proved to be the same as the efficiency of the corresponding rank score tests relative to the $t$-test in the two sample problem.